Skip to main content
SpringerLink
Log in

Two-dimensional Dirac oscillator in a magnetic field in deformed phase space with minimal-length uncertainty relations

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the dynamics of the Dirac oscillator in a magnetic field. The Heisenberg algebra is constructed in detail in the noncommutative phase space in the presence of minimal length. By means of the Nikiforov–Uvarov method, the energy eigenvalues are obtained exactly and the corresponding wave functions, in momentum space, are expressed in terms of hypergeometric functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Moreno and A. Zentella, “Covariance, CPT and the Foldy–Wouthuysen transformation for the Dirac oscillator,” J. Phys. A: Math. Gen., 22, L821–L825 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  2. J. Bentez, R. P. Martínez y Romero, H. N. Núez-Yépez, and A. L. Salas-Brito, “Solution and hidden supersymmetry of a Dirac oscillator,” Phys. Rev. Lett., 64, 1643–1645 (1990); Erratum, 65, 2085–2085 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. V. I. Kukulin, G. Loyola, and M. Moshinsky, “A Dirac equation with an oscillator potential and spin-orbit coupling,” Phys. Lett. A, 158, 19–22 (1991).

    Article  ADS  MathSciNet  Google Scholar 

  4. D. Itô, K. Mori, and E. Carriere, “An example of dynamical systems with linear trajectory,” Nuovo Cimento A, 51, 1119–1121 (1967).

    Article  ADS  Google Scholar 

  5. M. Moshinsky and A. Szczepaniak, “The Dirac oscillator,” J. Phys. A: Math. Gen., 22, L817–L819 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  6. M. R. Setare and O. Hatami, “Exact solutions of the Dirac equation for an electron in a magnetic field with shape invariant method,” Chin. Phys. Lett., 25, 3848–3851 (2008).

    Article  ADS  Google Scholar 

  7. M. Bednar, J. Ndimubandi, and A. G. Nikitin, “On connection between the two-body Dirac oscillator and Kemmer oscillators,” Canadian J. Phys., 75, 283–290 (1997).

    Article  ADS  Google Scholar 

  8. C. Quesne and M. Moshinsky, “Symmetry Lie algebra of the Dirac oscillator,” J. Phys. A: Math. Gen., 23, 2263–2272 (1990).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. J. Beckers and N. Debergh, “Supersymmetry, Foldy–Wouthuysen transformations, and relativistic oscillators,” Phys. Rev. D, 42, 1255–1259 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  10. M. Moshinsky, G. Loyola, and C. Villegas, “Anomalous basis for representations of the Poincaré group,” J. Math. Phys., 32, 373–381 (1991).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. M. Moshinsky and G. Loyola, “Mass spectra of the particle-antiparticle system with a Dirac oscillator interaction,” in: Workshop on Harmonic Oscillators (University of Maryland, College Park, Maryland, March 25–28, 1992, NASA Conference Publication Series, Vol. 3197, D. Han, Y. S. Kim, and W. W. Zachary, eds.), NASA, Washington (1993), pp. 405–421.

    Google Scholar 

  12. A. Del Sol Mesa and M. Moshinsky, “Relations between different approaches to the relativistic two-body problem,” J. Phys. A: Math. Gen., 27, 4685–4693 (1994).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. J. Beckers, N. Debergh, and A. G. Nikitin, “On supersymmetries in nonrelativistic quantum mechanics,” J. Math. Phys., 33, 152–160 (1992); arXiv: math-ph/0508021.

    Article  ADS  MathSciNet  Google Scholar 

  14. N. Debergh, J. Ndimubandi, and D. Strivay, “On relativistic scalar and vector mesons with harmonic oscillatorlike interactions,” Z. Phys. C, 56, 421–425 (1992).

    Article  ADS  Google Scholar 

  15. V. V. Dvoeglazov, “The Dirac–Dowker oscillator,” Nouvo Cimento A, 107, 1785–1788 (1994); arXiv: hep-th/9404145.

    Article  ADS  Google Scholar 

  16. M. Hosseinpour, H. Hassanabadi, and M. de Montigny, “The Dirac oscillator in a spinning cosmic string spacetime,” Eur. Phys. J. C, 79, 311, 7 pp. (2019); arXiv: 1904.05889.

    Article  ADS  Google Scholar 

  17. F. A. Dossa and G. Y. H. Avossevou, “Relativistic dynamics for a particle carrying a non-Abelian charge in a non-Abelian background electromagnetic field,” J. Math. Phys., 61, 022302, 13 pp. (2020).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. S. Capozziello, G. Lambiase, and G. Scarpetta, “Generalized uncertainty principle from quantum geometry,” Internat. J. Theor. Phys., 39, 15–22 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Kempf, G. Mangano, and R. B. Mann, “Hilbert space representation of the minimal length uncertainty relation,” Phys. Rev. D, 52, 1108–1118 (1995); arXiv: hep-th/9412167.

    Article  ADS  MathSciNet  Google Scholar 

  20. L. N. Chang, D. M. Minic, N. Okamura, and T. Takeuchi, “Effect of the minimal length uncertainty relation on the density of states and the cosmological constant problem,” Phys. Rev. D, 65, 125028, 8 pp. (2002); arXiv: hep-th/0201017.

    Article  ADS  Google Scholar 

  21. S. Benczik, L. N. Chang, D. Minic, N. Okamura, S. Rayyan, and T. Takeuchi, “Short distance versus long distance physics: The classical limit of the minimal length uncertainty relation,” Phys. Rev. D, 66, 026003, 11 pp. (2002); arXiv: hep-th/0204049.

    Article  ADS  Google Scholar 

  22. F. Brau, “Minimal length uncertainty relation and the hydrogen atom,” J. Phys. A: Math. Gen., 32, 7691–7696 (1999); arXiv: quant-ph/9905033.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. R. Akhoury and Y.-P. Yao, “Minimal length uncertainty relation and the hydrogen spectrum,” Phys. Lett. B, 572, 37–42 (2003); arXiv: hep-ph/0302108.

    Article  ADS  MATH  Google Scholar 

  24. F. A. Dossa, “One-dimensional harmonic oscillator problem and its hidden \(SU(1,1)\) symmetry in the presence of a minimal length,” Phys. Lett. A, 384, 126891, 8 pp. (2020).

    Article  MathSciNet  MATH  Google Scholar 

  25. F. A. Dossa, “Thermodynamic properties and algebraic solution of the \(N\)-dimensional harmonic oscillator with minimal length uncertainty relations,” Phys. Scr., 96, 105703, 10 pp. (2021).

    Article  ADS  Google Scholar 

  26. K. Nouicer, “An exact solution of the one-dimensional Dirac oscillator in the presence of minimal lengths,” J. Phys. A: Math. Gen., 39, 5125–5134 (2006).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. C. Quesne and V. M. Tkachuk, “Dirac oscillator with nonzero minimal uncertainty in position,” J. Phys. A: Math. Gen., 38, 1747–1765 (2005); arXiv: math-ph/0412052.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Z. Selema and A. Boumal, “Two-dimensional boson oscillator under a magnetic field in the presence of a minimal length in the non-commutative space,” Rev. Mex. Fis., 67, 226–237 (2021).

    Article  Google Scholar 

  29. A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics. A Unified Introduction with Applications, Birkhäuser, Basel (1988).

    MATH  Google Scholar 

  30. A. Kempf and G. Mangano, “Minimal length uncertainty relation and ultraviolet regularization,” Phys. Rev. D, 55, 7909–7920 (1997); arXiv: hep-th/9612084.

    Article  ADS  Google Scholar 

  31. S. Hossenfelder, “A note on theories with a minimal length,” Class. Quantum Grav., 23, 1815–1821 (2006); arXiv: hep-th/0510245.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. U. Harbach, S. Hossenfelder, M. Bleicher, and H. Stoecker, “Signatures of a minimal length scale in high precision experiments,” arXiv: hep-ph/0404205.

  33. U. Harbach and S. Hossenfelder, “The Casimir effect in the presence of a minimal length,” Phys. Lett. B, 632, 379–383 (2006); arXiv: hep-th/0502142.

    Article  ADS  Google Scholar 

  34. S. Hossenfelder, “The minimal length and large extra dimensions,” Modern Phys. Lett. A, 19, 2727–2744 (2006); arXiv: hep-ph/0410122.

    Article  ADS  MATH  Google Scholar 

  35. K. Konishi, G. Paffuti, and P. Provero, “Minimum physical length and the generalized uncertainty principle in string theory,” Phys. Lett. B, 234, 276–284 (1990).

    Article  ADS  MathSciNet  Google Scholar 

  36. E. Sadurní, “The Dirac–Moshinsky oscillator: theory and applications,” AIP Conf. Proc., 1334, 249–290 (2011); arXiv: 1101.3011.

    Article  ADS  Google Scholar 

  37. B. Mirza and M. Mohadesi, “The Klein–Gordon and the Dirac oscillators in a noncommutative space,” Commun. Theor. Phys., 42, 664–668 (2004).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  38. A. Boumali and H. Hassanabadi, “The thermal properties of a two-dimensional Dirac oscillator under an external magnetic field,” Eur. Phys. J. Plus, 128, 124, 13 pp. (2013).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences et Techniques, Université Nationale des Sciences, Technologies, Ingénierie et Mathématiques d’Abomey, Abomey, Bénin

    F. A. Dossa

  2. Laboratoire de Recherche en Physique Théorique, Institut de Mathématiques et de Sciences Physiques, Université de Porto-Novo, Porto-Novo, Bénin

    J. T. Koumagnon, J. V. Hounguevou & G. Y. H. Avossevou

Authors
  1. F. A. Dossa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. T. Koumagnon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. V. Hounguevou
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. G. Y. H. Avossevou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. A. Dossa.

Ethics declarations

The authors declare no conflicts of interest.

Additional information

Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 213, pp. 495–504 https://doi.org/10.4213/tmf10282.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dossa, F.A., Koumagnon, J.T., Hounguevou, J.V. et al. Two-dimensional Dirac oscillator in a magnetic field in deformed phase space with minimal-length uncertainty relations. Theor Math Phys 213, 1738–1746 (2022). https://doi.org/10.1134/S0040577922120078

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577922120078

Keywords

Search

Navigation